Wheel Speed.

Progress, Volume IV, Issue 8, 1 June 1909, Page 260

Wheel Speed.

(To the Editor Progress.) SIR, — With reference to Mr. Poolman's query m PROGRESS, May Ist, " Do all parts of a locomotive wheel move at the same speed ? " I have a word. Consider points A, B at the extremities of a diameter A, B. Let A% be first position of point A, and let the locomotive axle X move (roll) through a distance so small that the circular measure of the angle

O= h. Then A assumes the position of A on the dotted 0, and the diameter, which must pass through X in both position, shows that B has moved through 2h, that is, that Bi -B 2 = 2(X J - X 2).X 2 ). Below is a sketch (Pig. 2) of the curve a particle on the Oe makes, and (Fig. 3) the curve of a particle intermediate between the centre and circumference. The curve of the axis X is of course a straight line.

I think this clearly shows that a particle of a wheel on the circumference at the top travels at a rate of 2S (where S = speed of engine), and at the hottom is stationary. — I am, etc. D. A. Crawford.

(To the Editor Progress.) Sir, — In reply to Mi\ H. W. Poolman's letter on page 244 of your issue published May Ist, I have to say that the top of the wheel of any vehicle always moves at exactly twice the rate at which the centre of the wheel is travelling If a circle, or wheel keeping always in the same plane, be made to roll along the right line AB until a fixed point P, in its circumference, which at first touched the line at A, touches it again at B, after a complete revolution, the curve APB described by the motion of the point P is called a cycloid. The properties of this curve were investigated in the 17th century. We will suppose the circle, or wheel, to be 12 feet in circum-

ferenee, and tbe point P at A, and the circle to roll towards B, the centre C moving at a uniform rate, in a horizontal direction. When the wheel moves on a distance of one foot, the bottom will be at 1 on the line, and the point P at 1 on the cycloid; when the bottom is at 2 on thie line, P will be at 2 on the cycloid, and so on. A

glance at the diagram will show that the distance on the cycloid from 5-6 is much greater than fiom 5-6 on the line below. These distances can be exactly calculated; and the whole length of the cycloid from A to B is 4 times the diameter of the circle, or a little over 15 % feet. The following are the advances in a hoiizontal direction of the point P (in feet and decimals) for each successive foot that the centre advances: — A—l=.Os;A — l=.O5; 1— 2=.30; 2— 3=.74; 3—4=1.26; 4—5=1.70; 5—6=1.95. The total of these is 6 feet, which brings the point P to the top of The circle, as slpwn in the diagram. So much foi the top of the ciicle. The bottom of the circle, or wheel, is stationary. I suppose some people will object to

this statement, but it is true, nevertheless, and it may be proved as follows: — Any point P on the circumference moves onward and upward fiom A to 6, and onward and downward from 6 to B; then it rises again, and so on Now, if a point falls to a point B and then lises again, it must stop before it can rise again, and it is at that instant, between its falling and rising again, that it is stationary. As soon as it begins to move it is no longer thye bottom of the wheel. But the top of the wheel travels continuously at twice the speed of the centre This can be proved by taking the inciements 1, 2, 3, &c. indefinitely small (instead of one foot), when it will be found that one increment at the top will be double one at the centre. We are, etc., The Adams Engineering Co.